## E m811

These expressions are the result of the Lorentz transformations [Einstein, 1916; Lang, 1999; Froning, 1983] introduced by Einstein because [Harwit, 1973, Chapter 5] they allow both the laws of mechanics and those of electromagnetism to stay the same when changing inertial frame of reference (unlike the laws of dynamics, the Maxwell equations of electromagnetism change when classical Galilean transformations are used to correlate inertial frames). In 1948 Einstein discarded the concept of relativistic mass defined by equation (8.10) in favor of relativistic energy (8.11) only, which is completely consistent with the four-dimensional momentum formulation of his original theory (see [Miller, 1981] for details).

Inspection of equation (8.11) shows that, over long trips at sustained power such that spacecraft speed starts approaching the speed of light, there appears a new problem. In Newtonian mechanics applying a thrust F to a mass M results in an acceleration F/M. The thrust power needed is FV, growing with V 3 if V is the velocity of the mass ejected. Power stays always finite. At high V/c instead, the relativistic equation (8.11) predicts that more and more energy is needed as V/c grows, tending to infinity as V approaches light speed. Because energy can be produced only by mass conversion, the implication is that to reach higher and higher speed the mass to carry would have to be larger and larger. In the end, to achieve light speed the energy required is infinite, meaning the mass to be converted into energy would also be infinitely large. Thus, following the question of power, the second question is, how much mass will be needed to accelerate a spacecraft when the energy required increases faster and faster with spacecraft speed? This question can be better posed in terms of the ratio between initial and final spacecraft mass, the mass ratio (or weight ratio) MR. This ratio must be reasonable, and the Tsiolkowski law suggests that, to keep it so, the propulsion system must be capable of Isp much higher than today's, perhaps by a factor 102 to 103.

Figure 8.4 tells that only fusion rockets, or their limit case of matter-antimatter annihilation rockets, could theoretically reach such Isp. For the 10-ton spacecraft previously considered in Chapter 1 the LEO weight is 1,000 tons (2,205,000 lb). That is less than some large vertical launch rocket launchers, that have lift-off mass order of 2,000 tons (4,410,000 lb). Such mass is significant to put into orbit, but an Energia class launcher with a 230-ton cargo capability could lift it in five launches. If the 300-ton configuration were used with a tandem payload section, instead of a laterally mounted cargo container, then only four launches would be necessary.

In reality, a 10-ton payload for such a mission is insufficient. For long duration at least a 100-ton spacecraft is necessary and the launch weight from LEO for a oneway mission is now 10,000 tons (22,050,000 lb). The results would be a massive vehicle in LEO, perhaps such as the one artistically illustrated in Figure 8.5. As propellant tanks empty, they would be discarded to reduce the empty weight of the spacecraft and therefore reduce the propellant consumed. For this duration, the ship would have to have an energy source that could sustain thrust over the duration required. At this point, the only such energy source with the Isp needed is based on fusion or antimatter annihilation, and the ideal mission time, tmission, would be determined by the fact that the average thrust power

is related to the potential energy available onboard

by the constraint that

The time and distance permitted by a particular propulsion system and mass ratio are not strictly related to whether the spacecraft is manned or robotic. But the assets required to sustain conscious human beings over long durations (perhaps 10 to 20 years) result in a prohibitive weight and volume penalty. For such a mission, a future spacecraft would have to be a self-sufficient, integrated ecological support system. In this chapter only unmanned, robotic missions are considered in the determination of size and weight of spacecraft with respect to different propulsion systems.

To operate a propulsion system when speed approaches a significant fraction of the speed of light, energy and mass must be treated relativistically, and the constant acceleration strategy valid for exploring the Solar System may no longer be a template for stellar trips. The constraint V/c < 1 affects all aspects of spacecraft, including that of its propulsion system. For fast QI and interstellar travel the Isp (or, exhaust Ve) must be much higher than ever thought possible in the past and become no longer negligible with respect to c. This means that gas-dynamics and magneto-hydro-dynamics (MHD) should be reformulated to account for relativistic effects inside the propulsion systems themselves. Although relativistic equations of motion for gases and plasmas have been developed, they are far from having been universally accepted, let alone understood, for application to realizable propulsion systems (e.g., see [Anile and Choquet-Bruhat, 1987]).

This is a strong caveat, suggesting that issues associated with relativistic propulsion systems be left aside, at least insofar as they are based on the principle of action and reaction. The analysis that follows will assume Ve/c sufficiently smaller that relativistic effects may be neglected, and will examine what propulsion systems, if any, are likely to work over interstellar or quasi-interstellar distances. Energy density and power are some of the key aspects in answering these questions.

It is also understood that theoretical considerations, for instance, about fusion and its implementation in a rocket, are solidly grounded in established physics, but that true propulsion applications do not exist yet. Therefore many if not all of the systems discussed or outlined, and all of the most innovative concepts, are speculative.

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